Retrieval of Chaotic structure along with P-test to Truncated M-fractional Paraxial wave model

The Painlevé analysis has attracted considerable attention in the age of mathematical and physical sciences as a crucial method for analyzing FNLPDEs. This methodology is an effective way to assess if a given differential equation ensures the Painlevé property. The concept of the Painlevé property and its connection to integrability has profoundly understood several dynamical systems, including nonlinear processes and classical and quantum mechanics. This work aims to investigate the fundamental ideas behind the Painlevé test, its applications in several scientific fields, and its role in bifurcation and chaos theory in the study of dynamical system behavior. Understanding difficult mathematical and physical concepts, highlighting its function as a cornerstone in modern scientific research.